Is it just mischief making? Surely Diane Abbott doesn't have a chance of winning the Labour leadership. She can't play football. You have play football with the blokes to get on in Labour.

Then again, it was unlikely she'd get on the ballot paper, but she did. And as she said in the New Statesman debate, it was unlikely that she would get to Cambridge, but she did. And most unlikely of all that she'd get to Parliament, and she did that too.

And remember how unlikely it was that David Cameron would win the leadership of the Conservative party? Or how unlikely that Nick Clegg would get into government. Such things do come to pass. Life is 10-1 against.

But how could a female, black, rebellious left-winger appeal to middling England from her deprived constituency? How could she relate to white, property-owning, ambitious Britain? It's one of those lefty arguments that experience doesn't always support. For instance, those well-off, upper-middle class, public school educated millionaires – their background hasn't prevented them from talking to the country.

This is not to say class resentment doesn't exist. But stronger than the plumbers-and-butlers thing that Labour likes to fight on, is the disenchantment and the polite dislike the British public feels for its political class. Their language. Their manners. Their calculations.

Miss Abbott – as her answering machine calls her – is still what you'd call more of a person than a politician. All the other candidates have had their views, values and positions shaped by office, by focus groups, by years of frontbench life.

The fact that she sent her son to a fee-paying school (having slagged off her new patron Harriet Harman for doing the same thing) has discredited her in the eyes of the left.

Her defence was original and honest. She said what she'd done was "indefensible". The format of a political apology of the time went: "If anyone has got the impression that I have been inconsistent then that is something obviously that I regret." But there she was. She had been "incoherent", "inconsistent" and "indefensible".

She said she'd thrown away her reputation for political consistency, "But I threw it away in the best possible cause" (her son). The left doesn't like that sort of talk – but it rings a bell with ambitious parents of every creed and colour. And she does have a majority of 14,408.

Now, diversity is a complex doctrine for amateurs to get to grips with. When Hugh Quarshie played Henry V, we weren't supposed to notice that he was a black man (so my tutor told me at the time). On the other hand, Diane Abbott's colour is important and doctrinally meant to advertise something positive about us and our sympathies. In fact, colour and ethnicity is a minor element of Abbott's appeal.

She's big, she's cheerful, she's brave, she's solid. She's eloquent. She has a natural grasp of oratory and conversational rhetoric. She says things in such a way that people like to listen to her. Her virtues come from her background and experience of life. Where three of her opponents were born into the upper reaches of their societies (Ed Balls even spent some time at Eton), her parents were working class immigrants, and she came up through a grammar school and the glory of Cambridge education. She doesn't have to triangulate, it's the factual story of her life.

She is unbowed by the whips. Her career in the Commons has been marked by many rebellions, and by no means all from the firebrand left.

In the last couple of years she voted against her government on control orders, the Fiscal Responsibility Bill, mandatory pay audits, welfare reform, coroners and justice, ID cards, Ghurka resettlement, Heathrow third runway, the Lisbon Treaty, police raid on MP's office, post charge questioning and of course 42 days – in the course of which she wrung the withers of everyone watching, causing her Tory follow-on to say: "I have the near impossible task of following one of the finest speeches I have heard since being elected to the House of Commons."

And in that vein, she was, by some accounts, the easy winner of the New Statesman debate – warm, fresh, forceful, funny. She's popular. She may be a bit bonkers, as some say, but it's in a likeable way – and she does carry the crowd.

The polls must be a cause for concern for Labour voters. PoliticsHome shows the popularity rating of the four male candidates. They're minus. Negative. The least unpopular is Andy Burnham, almost certainly because no one knows who he is. The most unpopular is, obviously, Ed Balls at -39 points.

They have had the burden of office to carry. They haven't been able to say of Iraq, or top-up fees, or 90 days detention without trial that these decisions or proposals were "inconsistent, incoherent, indefensible". They are trapped by their responsible past. But with those three front-running males, it is possible that the more the public sees of them, the less we will like them. The fraternal rivalry between the Milibands is going to blow one day, and it'll only need to happen once.

Doubtless, in the unlikely event of her winning she would be mired in contradictions herself. But as the vote goes to the wider party she will be a fresh voice, a lively presence, careless of convention and one who has survived the pieties of the day. People like these things.

"The real danger," a Guardian writer said this week, "is that Diane Abbot might actually win." He also said: "It is a foolish error to give a leg-up to someone whose policies would guarantee a Conservative government at the next election."

Policies? Let's leave them to one side. They are just the measures; what's important is the men. Or in this case, the woman.

Diane Abbott: The CV

Full name Diane Julie Abbott

Born 27 September 1953

Education Harrow County Grammar School for Girls and then Newnham College, Cambridge, where she read history.

Career before politics Administration trainee at the Home Office (1976 to 1978), race relations officer at the National Council for Civil Liberties (1978 to 1980). Researcher and reporter at Thames Television (1980 to1983), researcher and reporter at TV-AM (1983 to 1985). Press officer at the Greater London Council (1985 to 1986) and head of press and public relations at Lambeth Council (1986 to 1987).

Political career Member of Parliament for Hackney North and Stoke Newington since 1987, when she became the first black woman to be elected to the House of Commons. She became one of just four Members of Parliament from ethnic minorities.

Controversial moments A well-known left-wing rebel, Abbott rebelled against the Labour government on control orders, ID cards, Ghurka resettlement, the Heathrow third runway, the Lisbon Treaty, and 42 days detention. Her decision in 2003 to send her son to the private City of London School, was seen as hypocritical because she had previously criticised Tony Blair and Harriet Harman for sending their children to selective state schools. She herself admitted that her actions were "indefensible" and "intellectually incoherent".

Leadership Abbott announced on the Today programme that she would stand in the leadership contest. She complained that there was "little choice" between the other candidates, all white males. She secured the 33 nominations necessary to appear on the ballot paper only after the withdrawal of fellow left-wing candidate, John McDonnell, and a decision by David Miliband to get his supporters to nominate her.

Abbot has been a regular guest on the BBC's This Week, on the couch with her old schoolfriend, Michael Portillo. She has however recently stood down from the role for the duration of the leadership contest.